A higher-dimensional geometrical approach for the classification of 2D square-triangle-rhombus tilings

Square-triangle-rhombus ($\mathcal{STR}$) tilings are encountered in various self-organized multi-component systems. They exhibit a rich structural diversity, encompassing both periodic tilings and long-range ordered quasicrystals, depending on the proportions of the three tiles and their orientation distributions. We derive a general scheme for characterizing $\mathcal{STR}$ tilings based on their lift into a four-dimensional hyperspace. In this approach, the average hyperslope ($2 \times 2$) matrix $\mathcal{H}$ of a patch defines its global composition with four real coefficients: $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$, and $\mathcal{W}$. The matrix $\mathcal{H}$ can be computed either directly from the area-weighted average of the hyperslopes of individual tiles or indirectly from the border of the patch alone. The coefficient $\mathcal{W}$ plays a special role as it depends solely on the rhombus tiles and encapsulates a topological charge, which remains invariant upon local reconstructions in the tiling. For instance, a square can transform into a pair of rhombuses with opposite topological charges, giving rise to local modes with five degrees of freedom. We exemplify this classification scheme for $\mathcal{STR}$ tilings through its application to experimental structures observed in two-dimensional Ba-Ti-O films on metal substrates, demonstrating the hyperslope matrix $\mathcal{H}$ as a precise tool for structural analysis and characterization.